Inspired by the novel preconditioning techniques developed for linear subproblems that appeared in nonlinear convex optimizations [1]}, [2]}, the new development of Kurdyka-Łojasiewicz (KL) analysis, and the proximal nonconvex optimization [3]}, [4]}, [5]}, [6]}, we proposed a preconditioned framework for the nonconvex alternating minimization aiming at dealing with the linear problems especially large-scale problems efficiently for (REF ).
Our contributions belong to
the following folds. First, we proposed a preconditioned framework that can deal with any linear subequation of \(u\)  or \(y\) . Any finite number of preconditioned iterations can guarantee the convergence of the whole nonlinear alternating minimization algorithm without solving the linear subequations with middle or high accuracy, which is different from the inexact solvers through error control. Especially, for the Ambrosio-Tortorelli model, preconditioned iterations can be employed for both linear equations involving \(u\)  and \(y\) . Second, with the analysis of the semialgebraic sets and o-minimal structure, we prove that all the functions \(L(u,y)\)  of discussed models are KL functions. Together with the boundedness of the iteration sequence, we obtain the global convergence of the proposed preconditioned alternating minimization by the recent developments of proximal minimization algorithms [3]}, [4]}, [5]}, [6]}. Third, our preconditioned framework can deal with the nonsmooth and nonconvex truncated quadratic problems efficiently with global convergence, while they are precluded by convex assumptions as in [11]}, [12]}, [13]} or \(C^1\)  conditions as in [14]}. Fourth, we developed several efficient red-black Gauss-Seidel preconditioners for both isotropic and anisotropic equations in divergence form by finite difference method [15]}, [16]}. Numerical tests show that one can get lower energy and better reconstructions more efficiently with the proposed preconditioned framework compared with solving the linear system of () with moderate accuracy and without any proximal terms.
